List of top Mathematics Questions asked in JEE Main

If a function f(x) defined by
\(f(x) = \begin{cases} ae^{x} + be^{-x}, & -1 \le x \lt 1 \\ cx^2, & 1 \le x \le 3 \\ ax^2 + 2cx, & 3 \lt x \le 4 \end{cases}\)
be continuous for some \(a, b, c \in R\) and \(f'(0) + f'(2) = e\), then the value of \(a\) is :

Let \(y = y(x)\) be the solution of the differential equation,
\(\frac{2 + \sin x}{y + 1}\cdot\frac{dy}{dx} = -\cos x\),
\(y \gt 0\), \(y(0) = 1\).
If \(y(\pi) = a\) and \(\frac{dy}{dx}\) at \(x = \pi\) is b, then the ordered pair (a, b) is equal to :

Let \(\alpha\) and \(\beta\) be the roots of the equation
\(5x^2 + 6x - 2 = 0\). If \(S_n = \alpha^n + \beta^n\), \(n = 1, 2, 3, \ldots\), then :

Area (in sq. units) of the region outside
\(\frac{|x|}{2} + \frac{|y|}{3} = 1\)
and inside the ellipse
\(\frac{x^2}{4} + \frac{y^2}{9} = 1\)
is :

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :

If the distance between the plane, \(23x - 10y - 2z + 48 = 0\) and the plane containing the lines
\(\frac{x + 1}{2} = \frac{y - 3}{4} = \frac{z + 1}{3}\)
and
\(\frac{x + 3}{2} = \frac{y + 2}{6} = \frac{z - 1}{\lambda}\) \((\lambda \in R)\)
is equal to \(\frac{k}{\sqrt{633}}\), then k is equal to ______.

If \(C_r \equiv {}^{25}C_r\) and \(C_0 + 5 \cdot C_1 + 9 \cdot C_2 + \ldots + (101) \cdot C_{25} = 2^{25} \cdot k\), then k is equal to ______.

Let \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) be three vectors such that \(|\vec{a}| = \sqrt{3}\), \(|\vec{b}| = 5\), \(\vec{b} \cdot \vec{c} = 10\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\frac{\pi}{3}\). If \(\vec{a}\) is perpendicular to the vector \(\vec{b} \times \vec{c}\), then \(|\vec{a} \times (\vec{b} \times \vec{c})|\) is equal to ______.

If the curves, \(x^2 - 6x + y^2 + 8 = 0\) and \(x^2 - 8y + y^2 + 16 - k = 0\), \((k \gt 0)\) touch each other at a point, then the largest value of k is ______.

If \(A = \{x \in \mathbb{R} : |x| \lt 2\}\) and \(B = \{x \in \mathbb{R} : |x - 2| \ge 3\}\); then :

If \(p \rightarrow (p \wedge \sim q)\) is false, then the truth values of p and q are respectively :

The number of terms common to the two A.P.'s \(3, 7, 11, \ldots, 407\) and \(2, 9, 16, \ldots, 709\) is ______.

The length of the minor axis (along y-axis) of an ellipse in the standard form is \(\frac{4}{\sqrt{3}}\). If this ellipse touches the line, \(x + 6y = 8\); then its eccentricity is :

Given : \(f(x) = \begin{cases} x, & 0 \le x \lt \frac{1}{2} \\ \frac{1}{2}, & x = \frac{1}{2} \\ 1 - x, & \frac{1}{2} \lt x \le 1 \end{cases}\) and \(g(x) = \left(x - \frac{1}{2}\right)^2,\ x \in R\).
Then the area (in sq. units) of the region bounded by the curves, \(y = f(x)\) and \(y = g(x)\) between the lines, \(2x = 1\) and \(2x = \sqrt{3}\), is :

A random variable X has the following probability distribution :

X:	1	2	3	4	5
P(X):	\(K^2\)	\(2K\)	\(K\)	\(2K\)	\(5K^2\)

Then \(P(X \gt 2)\) is equal to :

If \(\int \frac{d\theta}{\cos^2\theta\,(\tan 2\theta + \sec 2\theta)} = \lambda \tan\theta + 2\log_e |f(\theta)| + C\) where C is a constant of integration, then the ordered pair \((\lambda, f(\theta))\) is equal to :

Let [t] denote the greatest integer \(\le t\) and \(\lim_{x \to 0} x\left[\frac{4}{x}\right] = A\).
Then the function, \(f(x) = [x^2]\sin(\pi x)\) is discontinuous, when x is equal to :

If \(x = \sum_{n=0}^{\infty} (-1)^n \tan^{2n}\theta\) and \(y = \sum_{n=0}^{\infty} \cos^{2n}\theta\) for \(0 \lt \theta \lt \frac{\pi}{4}\), then :

If \(x = 2\sin\theta - \sin 2\theta\) and \(y = 2\cos\theta - \cos 2\theta\), \(\theta \in [0, 2\pi]\), then \(\frac{d^2y}{dx^2}\) at \(\theta = \pi\) is :

Let f and g be differentiable functions on R such that fog is the identity function. If for some a, b \(\in\) R, \(g'(a) = 5\) and \(g(a) = b\), then \(f'(b)\) is equal to :

Let \(a - 2b + c = 1\).
If \(f(x) = \begin{vmatrix} x + a & x + 2 & x + 1 \\ x + b & x + 3 & x + 2 \\ x + c & x + 4 & x + 3 \end{vmatrix}\), then:

The following system of linear equations
\(7x + 6y - 2z = 0\)
\(3x + 4y + 2z = 0\)
\(x - 2y - 6z = 0\), has

In the expansion of \(\left(\frac{x}{\cos\theta} + \frac{1}{x\sin\theta}\right)^{16}\), if \(\ell_1\) is the least value of the term independent of x when \(\frac{\pi}{8} \le \theta \le \frac{\pi}{4}\) and \(\ell_2\) is the least value of the term independent of x when \(\frac{\pi}{16} \le \theta \le \frac{\pi}{8}\), then the ratio \(\ell_2 : \ell_1\) is equal to :

Let a function \(f : [0, 5] \to R\) be continuous, \(f(1) = 3\) and F be defined as :
\(F(x) = \int_1^x t^2 g(t)\,dt\), where \(g(t) = \int_1^t f(u)\,du\).
Then for the function F, the point x = 1 is :

Let a, b \(\in\) R, a \(\ne\) 0 be such that the equation, \(ax^2 - 2bx + 5 = 0\) has a repeated root \(\alpha\), which is also a root of the equation, \(x^2 - 2bx - 10 = 0\). If \(\beta\) is the other root of this equation, then \(\alpha^2 + \beta^2\) is equal to :